Borel Set - Standard Borel Spaces and Kuratowski Theorems

Standard Borel Spaces and Kuratowski Theorems

The following is one of a number of theorems of Kuratowski on Borel spaces: A Borel space is just another name for a set equipped with a distinguished σ-algebra; by extension elements of the distinguished σ-algebra are called Borel sets. Borel spaces form a category in which the maps are Borel measurable mappings between Borel spaces, where

is Borel measurable means that f − 1(B) is Borel in X for any Borel subset B of Y.

Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines the topology of X and which makes X a complete separable metric space. Then X as a Borel space is isomorphic to one of (1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharam's theorem.)

Considered as Borel spaces, the real line R and the union of R with a countable set are isomorphic.

A standard Borel space is the Borel space associated to a Polish space.

Note that any standard Borel space is defined (up to isomorphism) by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum.

For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. See analytic set.

Every probability measure on a standard Borel space turns it into a standard probability space.